2015 AMC 12B Problem 22

First imagine everyone moves to the chair directly opposite. The condition becomes: each person must sit in the same chair or an adjacent one. The number of people who keep their seat must be even (otherwise an odd-length gap cannot be filled). If $0$ keep their seat, everyone shifts left, shifts right, or swaps with a neighbor: $4$ ways. If $2$ keep their seats, those two must be opposite or adjacent, giving $3+6=9$ choices, and the remaining people are forced to swap in adjacent pairs. If $4$ keep their seats, the other two must occupy adjacent seats and swap, giving $6$ choices. If all $6$ stay, there is $1$ way. The total is $4+9+6+1=20.$ Thus, the correct answer is D.